How To Actually Make Parallel Lines Intersect

How can parallel lines intersect if they are by definition supposed to maintain a constant width? This very notion sounds absurd. Yet, it is not only practically possible, but is also a necessity in mathematics and science.

In this essay, we will start with the historical development of the science behind optics and discover how this flows into the notion of intersecting parallel lines. Following this, we will explore practical applications of this concept as well. Let us begin.

This essay is supported by Generatebg

How Does Vision Work?

The notion of light rays bouncing off of objects and registering inside our eyes is a relatively recent one. What’s more, this is NOT a very intuitive concept. The proof for this is in the history pudding.

Many prominent people of science such as Plato and the Pythagoreans argued that sight is achieved by a fire-like phenomenon within our eyes. There were also other competing speculative theses about how our eyes produce light.

The theory of vision involving reflected light rays was first documented by Cairene mathematician Abu ‘Ali al-Hasan ibn al-Haytham in his optics book, Kitab al-Manazir.

Eventually, this book was translated into Latin. As a result, European people of science and art began developing a more sophisticated understanding of vision and perspective.

Art and Parallel Lines

There seems to have been a breakthrough in European art after the reflected light theory of vision had established itself. European artists were now able to paint more “realistic” paintings of what they observed in reality.

They began to understand that a point on their canvas represented a line in three-dimensional space. This meant that parallel lines from their landscape had to intersect on their canvas. If I’m losing you at this point, bear with me. We will clarify what is going on here next.

---

How to Actually Make Parallel Lines Intersect

Consider the following image. We know that the train tracks are parallel. However, they “appear” to meet at the horizon. Just what is going on here? Well, it turns out that the Renaissance painters had it figured.

Imagine that you are a Renaissance painter with a canvas and are trying to paint the above scene. From the reflected light theory of vision, you’d realise that every point in the 3D space in front of you forms a line with your eye (line of sight).

Let the rails be R1 and R2 respectively. Each point on these rails has a unique line of sight to your eyes. You will represent each of these rails using a line-like geometry on your canvas.

Here is the important realisation: similar to how each point on your canvas represents a line in the 3D space, each line in 3D space represents a plane on your canvas. So, if you sweep a plane through all the lines of sight from your eyes and rail R1, you end up with a plane P1. A similar process with R2 results in a plane P2.

When you are capturing the rails on your canvas, you are letting planes P1 and P2 cut your canvas. The rails R1 and R2 are parallel. However, the planes P1 and P2 are not, because they meet at your eye! Consequently, the lines L1 and L2 (representing the rails) are not parallel either.

They meet at a point V, which is known as the vanishing point.

---

The Art of Projection and More on the Vanishing Point

The phenomenon you just discovered as a Renaissance painter is known in mathematics as projective geometry. Almost all of us are so used to Euclidean geometry, that other geometry concepts feel counter-intuitive.

In projective geometry, parallel lines do intersect! To understand this, let us get back to the vanishing point. Not only do the lines L1 and L2 meet at V, but all the lines parallel to the tracks in reality must meet at V on your canvas.

There is only one type of lines that does not have a vanishing point on the canvas: lines that are parallel to the canvas itself (like the slats between the rails). These will remain parallel on the canvas as well.

There is actually nothing special about the train tracks. They just help us realise this phenomenon. There are more lines (of sight) through our eyes than there are points on the rails. Why? Because, there are horizontal lines from our eyes which don’t intersect on the ground at all. When you think about it, the term “horizon” seems to make practical sense now.

The vanishing point V, then, represents a point on the horizon that is infinitely far away from your position in the direction of the tracks. In mathematics, this is called a “point at infinity”. To understand this further, let us take a look at the projective plane.

The Projective Plane — Where Parallel Lines Intersect

The plane in the image below looks largely similar to the conventional two-dimensional plane that you are used to. However, note the subtle differences. Let us consider two points at infinity: P and Q.

If you travel in the positive y-direction infinitely, you will reach point P. But this is the same point that is infinitely far away from the origin in the negative y-direction as well. If you keep going along infinitely in the positive y-direction past P, you will be travelling along the negative y-axis towards the origin.

So, in a dramatic plot twist, the vertical axis appears to be a circle and not a line in the projective plane. The same is true for Q along the horizontal direction (x-axis). You must be thinking that this essay gets weirder the more you read. But wait, there’s more!

In Euclid’s plane, two points create a line. And two different lines meet at a single intersection point — unless they are parallel. If they are parallel, they never meet. However, in the projective plane, we discard the last condition. We just stick to the first two conditions. Consequently, projective plane geometry is governed by two axioms only:

1. Every pair of points creates a unique line.

2. Every pair of lines intersects exactly at one point.

There are no exceptions to these axioms, even if the lines are parallel to each other. Parallel lines just meet at infinity (think about the vanishing point). So, any line parallel to the vertical axis would meet at P as well. When you think about it, the notion of poles and lines of longitudes makes sense in this context.

---

Final Remarks

Jordon Ellenberg has the following to say about projective plane geometry:

“It’s perfectly symmetrical and elegant in a way that classical plane geometry is not.”

— Jordon Ellenberg.

Usually, if an elegant concept presents itself in mathematics, scientists eventually find an application for it in practice. Sometimes, it is the other way around; practical scientific concepts have elegant representations in the mathematical world.

Projective plane geometry is not just useful for Renaissance painting, but also for mapping complex internet landscapes for web applications, etc.

In the grand scheme of things, it just turns out that if you are willing to look beyond Euclid’s plane, parallel lines do indeed intersect. In fact, we have most recently progressed greatly in science (relativity theory, for example) by looking beyond Euclidean geometry.

---

Reference and credit: Jordon Ellenberg.

If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you:

• Non-Euclidean Geometry: The Forgotten Story
• How To Really Understand Fractals?
• Can You Actually Solve This Viral Geometry Puzzle?

If you would like to support me as an author, consider contributing on Patreon.